We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha \beta}(x/\varepsilon)\partial_{x_j}u(x)+b_i^\alpha(x,u(x))\right)=b^\alpha(x,u(x)) \mbox{ for } x \in \Omega. $$ For small $\varepsilon>0$ we prove existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of $\|u_\varepsilon-u_0\|_\infty$ for $\varepsilon \to 0$. Our assumptions are, roughly speaking, as follows: The functions $a_{ij}^{\alpha \beta}$ are bounded, measurable and $\mathbb{Z}^2$-periodic, the functions $b_i^\alpha(\cdot,u)$ and $b^\alpha(\cdot,u)$ are bounded and measurable, the functions $b_i^\alpha(x,\cdot)$ and $b^\alpha(x,\cdot)$ are $C^1$-smooth, and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^2$. Neither global solution uniqueness is supposed nor growth restrictions of $b_i^\alpha(x,\cdot)$ or $b^\alpha(x,\cdot)$ nor higher regularity of $u_0$, and cross-diffusion is allowed. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence, local uniqueness and error estimates for singularly perturbed problems and and for homogenization problems.

We consider a periodic boundary value problem for a singularly perturbed reaction-advection-diffusion equation in the case of a two-dimensional space variable. We construct a new interior layer-type formal asymptotics which includes an approximation of the location of the interior layer, investigate the order-preserving properties of the operators generating the asymptotics, and propose a modified procedure to obtain asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. We also prove the asymptotic Lyapunov stability of this solution. DOI 10.1134/S1061920824040113

Singularly perturbed systems of reaction-diffusion equations in the case of fast and slow equations, known as Tikhonov-type systems, are studied. The case where the roots of the degenerate equation intersect and hence are not isolated is considered. Effective conditions for the existence of a solution whose approximation is the so-called composite stable solution of the degenerate system are obtained. The existence of a solution is proved and the accuracy of the asymptotic approximation is estimated by using the asymptotic method of differential inequalities extended to this class of problems. Conditions for the Lyapunov stability of solutions such as those of the corresponding initial boundary value problems are given.

In the paper, the existence of a stable stationary solution in a reaction-diffusion system with slow and fast components in a two-dimensional spatial variable case is investigated. The theorem of the existence of a stationary solution with boundary layers in the case of Dirichlet boundary conditions is proven, its asymptotic approximation is constructed, and conditions for Lyapunov asymptotic stability of this solution are obtained. The research is based on the asymptotic method of differential inequalities, applied to a new class of problems. This result is practically important both for various applications described by similar systems and for the application of numerical stationing methods when solving elliptical boundary value problems.

В работе проведено исследование нового класса периодических по времени решений сингулярно возмущенных систем уравнений реакция-диффузия в случае быстрого и медленного уравнений, которые принято называть системами тихоновского типа. Построена погранслойная асимптотика решений, доказано существование решений с построенной асимптотикой, получены условия асимптотической устойчивости по Ляпунову таких решений как решений соответствующих начально-краевых задач. Библиография: 16 названий.

This paper studies time-periodic solutions of singularly perturbed Tikhonov systems of reaction–diffusion–advection equations with nonlinearities that include the square of the gradient of the unknown function (KPZ nonlinearities). The boundary layer asymptotics of solutions are constructed for Neumann and Dirichlet boundary conditions. The study considers both the case of quasimonotone sources and systems without the quasimonotonicity condition. The asymptotic method of differential inequalities is used to prove theorems on the existence of solutions and their Lyapunov asymptotic stability. DOI 10.1134/S1061920824030129

Исследуется существование стационарных решений сингулярно возмущенных систем уравнений реакция-диффузия-адвекция в случае быстрого и медленного уравнений реакция-диффузия-адвекция с нелинейностями, содержащими градиент искомой функции в квадрате (KPZ-нелинейностями). Для доказательства теорем существования используется асимптотический метод дифференциальных неравенств. Построена погранслойная асимптотика решений в случае граничных условий Неймана и Дирихле. Рассмотрен случай квазимонотонных источников и систем без требования квазимонотонности.

We study a new class of time-periodic solutions of singularly perturbed systems of reaction–diffusion equations in the case of a fast and a slow equation, which are usually called Tikhonov-type systems. A boundary layer asymptotics of solutions is constructed, the existence of solutions with this asymptotics is proved, and conditions for the Lyapunov asymptotic stability of these solutions treated as solutions of the corresponding initial–boudary value problems are obtained.

В работе проведено исследование сингулярно возмущенных систем уравнений реакция-диффузия в случае быстрого и медленного уравнений, которые принято называть системами тихоновского типа. Рассмотрен случай нарушения условия изолированности корней вырожденного уравнения вследствие пересечения корней вырожденного уравнения. Получены эффективные условия существования решения, приближением которого является так называемое составное устойчивое решение вырожденной системы. Доказательство существования решения и оценка точности асимптотического приближения получены с помощью распространенного на этот класс задач асимптотического метода дифференциальных неравенств. Получены условия устойчивости по Ляпунову решений, таких как решения соответствующих начально-краевых задач. Библиография: 20 названий.

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.

Развивается новый метод исследования неустойчивых контрастных структур - решений с внутренним переходным слоем, основанный на построении достаточно точных неупорядоченных верхних и нижних решений и применении следствия из теоремы Крейна-Рутмана. Сформулированы условия существования неустойчивых по Ляпунову одномерных контрастных структур типа ступеньки как стационарных решений сингулярно возмущенных параболических уравнений реакция-диффузия с разрывной правой частью. Показано, что полученные результаты можно распространить на другие сингулярно возмущенные одномерные задачи реакция-диффузия-адвекция с разрывными нелинейностями.

We consider direct and inverse problems of three-dimensional quasi-static elastography underlying a cancer diagnosis method. They are based on a model of a tissue exposed to surface compression with deformations obeying linear elasticity laws. The arising three-dimensional displacements of the tissue are described by a boundary value problem for partial differential equations with coefficients determined by a variable Young's modulus and a constant Poisson ratio. The problem contains a small parameter, so it can be solved using the theory of regular perturbations of partial differential equations. This is the direct problem. The inverse problem is to find the Young modulus distribution from given tissue displacements. A significant increase in Young's modulus within a certain tissue domain suggests possible malignancy. Under certain assumptions, simple formulas for solving both direct and inverse problems of three-dimensional quasi-static elastography are derived. Three-dimensional inverse test problems are solved numerically with the help of the proposed formulas. The resulting approximate solutions agree fairly well with the exact model solutions. The computations based on the formulas require only several tens of milliseconds on a moderate-performance personal computer for sufficiently fine grids, so the proposed small-parameter approach can be used in real-time cancer diagnosis.

We consider a boundary-value problem for singularly perturbed integro-differential equation describing stationary reaction–diffusion processes with due account of nonlocal interactions. The principal feature of the problem is the presence of a singularly perturbed Neumann condition describing intense flows on the boundary. We prove that there exists a boundary-layer solution, construct its asymptotic approximation, and establish its asymptotic Lyapunov stability. Illustrative examples are given.

An algorithm is presented for the construction of an asymptotic approximation of a stable stationary solution to a diffusion equation system in a two-dimensional domain with a smooth boundary and a source function that is discontinuous along some smooth curve lying entirely inside the domain. Each of the equations contains a small parameter as a factor in front of the Laplace operator, and as a result, the system is singularly perturbed. In the vicinity of the curve, the solution of the system has a large gradient. Such a problem statement is used in the model of urban development in metropolitan areas. The discontinuity curves in this model are the boundaries of urban biocenoses or large water pools, which prevent the spread of urban development. The small parameter is the ratio of the city’s outskirts linear size to the whole metropolis linear size. The algorithm includes the construction of an asymptotic approximation to a solution with a large gradient at the media interface as well as the steps for obtaining the existence conditions. To prove the existence and stability theorems, we use the upper and lower solutions, which are constructed as modifications of the asymptotic approximation to the solution. The latter is constructed using the Vasil’yeva algorithm as an expansion of a small parameter exponent.

We consider periodic homogenization of boundary value problems for quasilinear second-order ODE systems in divergence form of the type a(x,x/ε,u(x),u'(x))'= f(x,x/ε,u(x),u'(x)) for x ∈ [0,1]. For small ε>0 we show existence of weak solutions u=u_ε as well as their local uniqueness for u-u_0_∞≈ 0, where u_0 is a given non-degenerate solution to the homogenized boundary value problem, and we describe the rate of convergence to zero for ε→ 0 of the homogenization error u_ε-u_0_∞. In particular, we show that this rate depends on the smoothness of the maps a(·,y,u,u') and f(·,y,u,u'). Our assumptions are, roughly speaking, as follows: The maps a,f:[0,1]×ℝ×ℝ^n×ℝ^n→ℝ^n are continuous, the maps a(x,y,·,·) and f(x,y,·,·) are C^1-smooth, the maps a(x,·,u,u') and f(x,·,u,u') are 1-periodic, and the maps a(x,y,u,·) are strongly monotone and Lipschitz continuous uniformly with respect to x, y and bounded u. No global solution uniqueness is supposed. Because x is one-dimensional, no correctors and no cell problems are needed. But, because the problem is nonlinear, we have to care about commutability of homogenization and linearization. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence and local uniqueness results for singularly perturbed problems and and for homogenization problems.

In the paper, a boundary value problem for a singularly perturbed reaction-diffusion-advection equation is considered in a two-dimensional domain in the case of discontinuous coefficients of reaction and advection, whose discontinuity occurs on a predetermined curve lying in the domain. It is shown that this problem has a solution with a sharp internal transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion in a small parameter is constructed, and also sufficient conditions are obtained for the input data of the problem under which the solution exists. The proof of the existence theorem is based on the asymptotic method of differential inequalities. It is also shown that a solution of this kind is Lyapunov asymptotically stable and locally unique. The results of the paper can be used to create mathematical models of physical phenomena at the interface between two media with different characteristics, as well as for the development of numerical-analytical methods for solving singularly perturbed problems.

We consider the initial-boundary value problem of reaction-diffusion-advection that has a solution of a front form. The statement comes from the theory of wave physics. We study the question of the solution stabilizing to the stationary one. Proof of the stabilization theorem is based on the concepts of upper and lower solutions and corollaries from comparison theorems. The upper and lower solutions with large gradients are constructed as modifications of the formal moving front asymptotic approximation in a small parameter. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary value problem get into the attraction domain of the asymptotically stable stationary solution on a sufficiently large time interval. The study conducted in this work gives an answer about the non-local attraction domain of the stationary solution and can give some stationing criteria. The results are illustrated by computational examples.

We consider families u = u(epsilon,0) of boundary layer solutions to singularly perturbed quasilinear problems of the type epsilon(2) (a(x, u(x), s)u' (x))V = b(x, u(x), epsilon) for x is an element of (-1, 1), u(-1) = u' (1) = 0, and we describe the behaviour of these solution families under small regular, but nonsmooth perturbations, i.e. we show existence and local uniqueness of solutions u = u(epsilon,delta) approximate to u(epsilon,0) to -epsilon(2) (a(x, u(x), epsilon)u'(x))' + b(x, u(x), epsilon) = delta g(x) for x is an element of (-1, 1), u(-1) = u' (1) = 0 with epsilon approximate to 0 and delta approximate to 0. Roughly speaking, we show the following: If g is a Dirac function, then for all smalls epsilon > 0 and delta >= 0, such that delta/epsilon is small, those solutions exist, and parallel to u(epsilon,delta) - u(epsilon,0)parallel to(infinity) = O(delta/epsilon) for delta -> 0 uniformly with respect to epsilon. If g is an element of L-2(-1, 1), then for all small epsilon > 0 and delta > 0, such that delta/root epsilon is small, those solutions exist, and parallel to u(epsilon,delta) - u(epsilon,0)parallel to(infinity) = O(delta/root epsilon) for delta -> 0 uniformly with respect to epsilon. And if g is an element of L-infinity (-1, 1), then for all smalls epsilon > 0 and delta >= 0 those solutions exist, and parallel to u(epsilon,delta) - u(epsilon,0)parallel to(infinity) = O(delta) for delta -> 0 uniformly with respect to epsilon. Finally we show that these asymptotic estimates are optimal. (C) 2023 Elsevier Inc. All rights reserved.

Рассмотрена сингулярно возмущенная периодическая задача для уравнения типа Бюргерса с модульной адвекцией и периодическим линейным усилением. Получены условия существования, единственности и асимптотической устойчивости по Ляпунову периодического решения с внутренним переходным слоем, построено его асимптотическое приближение. Асимптотика решения применена для определения граничных условий, обеспечивающих реализацию заданного режима движения фронта - задачи граничного управления. Сформулировано понятие асимптотического решения задачи граничного управления и получены достаточные условия существования требуемого периодического режима движения фронта.

We obtain an asymptotic approximation to a moving inner layer (front) solution of an initial–boundary value problem for a singularly perturbed parabolic reaction–advection–diffusion equation with small advection. We separately consider the case of a continuous source (the nonlinearity describing the interaction and reaction) and the case of a source discontinuity for a certain value of the unknown function, which arises in a number of topical applications. For either problem, an asymptotic approximation to the solution is constructed and existence and uniqueness theorems for such a solution are proved.